Real-rooted atlas

A357613 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(2\,n - 10\,n^{2} + 2\,n^{2} x + 12\,n^{3} - 8\,n^{3} x + 8\,n^{4} x\right)\,P_{n}(x) = \left(-6 - 12\,x - 6\,x^{2} + 45\,n + 82\,n x + 35\,n x^{2} - 108\,n^{2} - 165\,n^{2} x - 54\,n^{2} x^{2} - n^{2} x^{3} + 81\,n^{3} + 63\,n^{3} x - 15\,n^{3} x^{2} - n^{3} x^{3} + 54\,n^{4} x + 48\,n^{4} x^{2} + 2\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(-3\,x - 6\,x^{2} - 3\,x^{3} + 18\,n x + 32\,n x^{2} + 14\,n x^{3} - 27\,n^{2} x - 39\,n^{2} x^{2} - 13\,n^{2} x^{3} - n^{2} x^{4} - 18\,n^{3} x^{2} - 20\,n^{3} x^{3} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
215401873
38433631580815
44952640378020163509281
53003200203861031680115501512106375
6185641485123603604004002252256177664681221305
7116280108528031744444455360340340014414403153152745614018975

All 15 rows

Technical details
Node id
n-7fb3d7e99783adbd · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2dbb8918a8b74f8e

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       3,        4;
  2:      15,       40,        18;
  3:      84,      336,       315,        80;
  4:     495,     2640,      3780,      2016,       350;
  5:    3003,    20020,     38610,     31680,     11550,      1512;
  6:   18564,   148512,    360360,    400400,    225225,     61776,      6468;
  7:  116280,  1085280,   3174444,   4455360,   3403400,   1441440,    315315,    27456;
  8:  735471,  7845024,  26860680,  45581760,  44089500,  25395552,   8576568,  1555840,  115830;
  9: 4686825, 56241900, 220641300, 439321344, 514829700, 376049520, 172830840, 48372480, 7482618, 486200;
%F Formula
The row polynomials satisfy (12*n^3-10*n^2+2*n+(8*n^4-8*n^3+2*n^2)*x)*P(n,x) = (81*n^3-108*n^2+45*n-6+(54*n^4+63*n^3-165*n^2+82*n-12)*x+(48*n^4-15*n^3-54*n^2+35*n-6)*x^2+(2*n^4-n^3-n^2)*x^3)*P(n-1,x) - ((27*n^2-18*n+3)*x+(18*n^3+39*n^2-32*n+6)*x^2+(20*n^3+13*n^2-14*n+3)*x^3+(2*n^3+n^2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 63*n^3 - 165*n^2 + 82*n - 12)*x + (48*n^4 - 15*n^3 - 54*n^2 + 35*n - 6)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*P[n - 1, x] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 39*n^2 - 32*n + 6)*x^2 + (20*n^3 + 13*n^2 - 14*n + 3)*x^3 + (2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(12*n^3 - 10*n^2 + 2*n + (8*n^4 - 8*n^3 + 2*n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((81*n^3 - 108*n^2 + 45*n - 6 + (54*n^4 + 63*n^3 - 165*n^2 + 82*n - 12)*x + (48*n^4 - 15*n^3 - 54*n^2 + 35*n - 6)*x^2 + (2*n^4 - n^3 - n^2)*x^3)*v[i - 1] - ((27*n^2 - 18*n + 3)*x + (18*n^3 + 39*n^2 - 32*n + 6)*x^2 + (20*n^3 + 13*n^2 - 14*n + 3)*x^3 + (2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(12*n^3 - 10*n^2 + 2*n + (8*n^4 - 8*n^3 + 2*n^2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.